Wireless (Physical Layer) Network Coding with Limited Hierarchical

1

Wireless (Physical Layer) Network Coding with Limited Hierarchical Side-Information: Maximal Sum-Rates in 5-Node Butterfly Network Tomas Uricar, Bin Qian, Student Member, IEEE, Jan Sykora, Member, IEEE and Wai Ho Mow, Senior Member, IEEE

Abstract—Favourable characteristics of wireless channels (including the inherent broadcast and superposition nature) provide a fertile ground for the extension of conventional Network Coding (NC) principles to wireless communication networks. However, the research of emerging Wireless (Physical layer) Network Coding (WNC) techniques have already revealed several non-trivial research problems which do not appear in the conventional (wireline) NC systems, including the sensitivity to channel parametrization and challenging multi-source transmission synchronization. In this paper we uncover another significant research challenge typical for multi-node WNC systems. We show that the performance of contemporary WNC bi-directional relaying strategies is dominated by the availability of a specific Hierarchical Side Information (HSI), required for the successful decoding of desired information from hierarchical (WNC-coded) data streams. We analyse the impact of unreliable transmission of HSI on the performance of Wireless Butterfly Network (WBN) and we show that all state-of-the-art relaying strategies must be appropriately modified to avoid the deterioration of WBN performance in the limited HSI regime. Index Terms—Wireless (Physical Layer) Network Coding, WNC/PLNC, Butterfly Network, Relaying, Superposition Coding

I. I NTRODUCTION ORE than a decade has passed since Ahlswede et al. established the Network Coding (NC) theory [1] in their seminal paper [2]. The NC-based combining of data packets has truly brought a revolution in understanding of communication in multi-node networks, allowing to boost their performance far beyond the conventional (routing-based) solutions. Later on, in 2006, three groups of scientists independently introduced the basic idea of Wireless (Physical Layer) Network Coding (WNC) [3]–[5], which enabled an efficient utilization of favourable characteristics of wireless channels (including the inherent broadcast and superposition nature) together with a modified NC processing. However, while the primordial NC principles assume that each communication network comprises

M

Manuscript received October 1, 2013; revised February 18, 2014; accepted June 3, 2014. This work was partially conducted when the first author was visiting HKUST under the SENG International Internship Program. Work of T. Uricar and J. Sykora was supported by the FP7-ICT DIWINE project, by the Grant Agency of the Czech Technical University in Prague, grant SGS, and by MEYS of the Czech Republic, grant LD12062. T. Uricar and J. Sykora are with the Czech Technical University in Prague, Czech Republic (e-mail: {uricatom, Jan.Sykora}@fel.cvut.cz). B. Qian and W. H. Mow are with the Hong Kong University of Science and Technology, Hong Kong (e-mail: {bqian, eewhmow}@ust.hk.

Figure 1. Network.

Half-duplex communication in the 5-node Wireless Butterfly

a set of independent reliable channels, the nature of practical wireless channels is far from this assumption. Consequently, the research of WNC techniques have already revealed several non-trivial research problems which do not appear in the conventional (wireline) NC systems, including the sensitivity to channel parametrization [6]–[10] or challenging multisource transmission synchronization [11]–[13]. A. Background an Related Work In a broad sense, the term "WNC" refers to all communication strategies where destination nodes decode the required information from two specific “hierarchical” information flows [14]. While the first flow carries only some combination/function of source data, and hence it is sometimes denoted as "hierarchical" (see e.g. [15]), the second flow contains a complementary side information which is essential to extract the desired information from the hierarchical one. To emphasize the mutual relation between these two specific information flows, we use the term "Hierarchical SideInformation" (HSI) to denote the second (i.e. side information) stream1 . The availability of HSI at receiving node can be considered as a prerequisite for implementation of WNC processing in any wireless system. Remarkably, the majority of recent research results in the field of WNC aims on the 2-Way Relay Channel (2-WRC) scenario [6], [18]–[23], where perfect HSI can be always secured by storing the previously sent data in nodes’ 1 Note that other terms like Complementary Side Information (C-SI) [16] or self-information [17] are equivalently used in the literature to denote HSI.

2

buffers (see e.g. [14]). In more complex wireless networks, however, a respective receiver can obtain the required HSI only by overhearing the information broadcast from other nodes in its vicinity (see Fig. 1). Even though that this information can be overheard virtually at no cost (exploiting the broadcast property of wireless channels), the full (perfect) HSI cannot be always decoded due to an insufficient capacity of the corresponding wireless channel [24]–[26]. To mitigate this problem, transmission rates in the system must be either properly reduced to maintain the availability of perfect HSI, or a modification of WNC processing for limited/partially available HSI must be employed to guarantee a decodability of the desired information at all receiving nodes [24]–[27]. B. Goals of this Paper and Contribution As noted in the previous section, unreliable HSI channels (Fig. 1) may easily become a bottleneck of the system if only perfect HSI processing is employed. Fortunately, as we show in this paper, this limitation can be overcome in all state-ofthe-art relaying strategies by allowing to process only a specific part of the overheard source transmission as a "partial HSI". We follow the methodology presented in one of the WNC seminal papers [14] to extend the performance analysis from 2-WRC to WBN with generally limited HSI. We evaluate the maximal (theoretically achievable) sum-rates of reference WNC schemes and we show that the optimality of perfect/partial HSI processing depends not only on the particular WNC strategy, but also on the actual channel gains in the WBN system. Besides of this we emphasize also the fact that an unequal duration of communication steps in WBN (Fig. 1) must be allowed to achieve the best performance of the system. Even though that we do not design any new modulationcoding scheme in this paper, the presented informationtheoretic investigations can provide some valuable suggestions for a practical implementation of WNC processing in WBN. In addition, we introduce an extension of a Superposition Coding (SC) relaying scheme2 from [29], which not only provides a feasible adaptive scheme for WBN with arbitrary HSI channel quality, but also can be relatively easily extended to a practical modulation-coding scheme (as shown e.g. in [30, Section V.]). The rest of this paper is organized as follows. First, we introduce the system model and summarize all the required definitions and assumptions in Section II. Then, an overview of relaying strategies, including the comparison of their performance in WBN is provided in Section III. The SC-based relaying scheme is introduced in Section IV, together with a numerical optimization of its performance. The paper is concluded in Section V. II. S YMMETRIC W IRELESS B UTTERFLY N ETWORK In WBN, sources SA , SB sent data to their respective destinations DA , DB . Since direct links are not available between the intended source-destination pairs (SA → DA , SB → DB ), a help of an intermediate relay node R is required to enable successful communication. All half-duplex nodes share the 2 The

SC scheme is in principal similar to the Han-Kobayashi scheme [28].

available time-frequency resources in a non-orthogonal way, and consequently, each communication round is divided into two separate steps (see Fig. 1). In step I, sources SA , SB transmit their data messages dA , dB to the relay R. Due to the inherent broadcast nature of wireless channels, transmission of source SA (respectively SB ) is overheard by its “unintended" destination DB (respectively DA ) as HSI (see Fig. 1). Generally only limited (partial) information can be gathered from this observation at destinations, depending on the actual capacity of the corresponding HSI channel. A particular relay processing (including the particular form of the relay output) depends on the specific WNC strategy (more details are provided in the following sections). After processing the received information, the relay broadcasts the output (hierarchical) message dR to DA , DB in step II (see Fig. 1). We denote the relay output as “hierarchical signal", since it generally contains some function of individual source data. Finally, the desired data can be decoded at both DA , DB from the relay (hierarchical) signal received in step II and HSI3 overheard in step I (see e.g. [15]). A. Definitions and Assumptions A signal space representation of the signal transmitted from node K and observed at node L is: yKL [m] = hKL xK [m] + zL [m] ,

(1)

where xK [m] denotes the m-th complex constellation symbol transmitted from node K, hKL is the complex channel coefficient on KL channel and z [m] is the complex additive Gaussian noise C N (0, N0 ) . Complex valued vector is denoted by x. All complex channel coefficients of source-relay (hSA R , hSB R ), source-destination (i.e. HSI – hSA DB , hSB DA ) and relaydestination channels (hRDA , hRDB ) are assumed to be constant during the observation and perfectly known by all nodes. This allows us to adapt the transmission rates in the system to the actual channel conditions (allowing an unequal duration of steps I, II) and thus to evaluate the maximum sum-rate performance of analysed WNC strategies. Each node uses the same transmission power and transmitted symbols from all nodes are zero-mean with a power normalized to unity. Consequently, the Signal-to-Noise Ratio (SNR) of a particular link can be defined as: 2 h i j γi j = i, j ∈ {SA , SB , DA , DB , R} . (2) N0 For simplicity reasons we assume a symmetric WBN in this paper4 and hence the channel SNRs can be summarized as 3 Note again that HSI carries a complementary side information to the hierarchical signal, i.e. a side information which is necessary to acquire the particular source data from the hierarchical signal (see e.g. [16]). 4 The assumption of symmetric channels is introduced solely to reduce the number of varying parameters (channel gains) in the analysed model. In principal, the analysis can be relatively easily extended to the asymmetric case, however this would reduce the transparency of the presented (closedform) results.

3

(see also Fig. 1):

the 2-step strategies (AF, JDF, HDF) as

γSA R = γSB R = γ1 , γSA DB = γSB DA = γ2 , γRDA = γRDB = γ3 .

(3)

B. Sum-rate Performance We extend the WNC performance analysis from 2-WRC (as presented in the seminal paper [14]) to WBN with generally limited HSI. Similarly as in [14], [30] we assume that all channels have bandwidth normalized5 to 1 Hz and hence a rate up to C (γ ) = log2 (1 + γ ) [bit/s] can be reliably sent through a channel with SNR γ . Time can be expressed in a number of symbols and hence Nr bits are transmitted when N symbols are sent with a rate equal to r. We assume that codewords are sufficiently long, securing a zero error probability for the rates below the channel capacity (i.e. if r ≤ C). To compare the performance of various WNC relaying schemes in WBN with limited HSI, we evaluate the sumrate (equivalent to the bi-directional rate in 2-WRC [14]). We define the sum-rate as the total sum of bits successfully transmitted between the intended source-destination pairs (SA → DA , SB → DB ) during one communication round (step I & step II): Definition 1. (Sum-rate): Let, during one communication round of length N = NI + NII symbols, destination DA receives reliably |dA | bits from source SA and destination DB receives reliably |dB | bits from source SB . Then the sum-rate in WBN is given by: |dA | + |dB | , (4) Rsum = NI + NII where |d| denotes the number of bits in a binary message d and NI (respectively NII ) is a length of step I (respectively step II) in symbols. The main goal of the paper is the analysis of the impact of limited HSI on the performance of state-of-the-art WNC strategies. In particular, we analyse Amplify & Forward (AF), Joint Decode & Forward (JDF) and Hierarchical Decode & Forward (HDF) strategies6 (see [14], [16]–[18], [20], [32] and references therein for a 2-WRC implementation of these strategies). For the sake of comparison, we analyse also the 3-step Decode & Forward (DF) strategy [14], where step I (Fig. 1) is time-shared by both sources (orthogonal source transmissions). Due to the system symmetry (see Fig. 1) the maximum sumrate can be achieved when both sources transmit at the same rate, i.e. when rA = rB = r and hence (4) can be rewritten for 5 The bandwidth normalization allows to express the rates in bits/s [14]. Moreover, it makes also the the terms “rate" and “spectral efficiency" equivalent [30]. 6 Note that some other relaying schemes (Denoise & Forward (DNF) [4], [6] and Compute & Forward (CaF) [31]) are built on the principles similar to HDF. The basic idea of all these relaying schemes is based on a direct decoding of functions of user signals (linear combinations of lattice codewords are decoded in CaF). Hence, in all HDF/DNF/CaF strategies, separate user data are not decoded, and intermediate network nodes process only some (hierarchical) functions of source data (see e.g. [15]).

R2step sum =

2rNI NI (rA + rB ) = NI + NII NI + NII

(5)

and similarly as 1 2 NI (rA + rB )

rNI , (6) NI + NII NI + NII for the 3-step strategy (DF), where equal time-sharing of step I is applied7. Note that NII , i.e. the length of step II generally depends on the length of step I (NI ), source rates (r), channel SNRs (γ1 , γ2 , γ3 ) and also on the employed WNC strategy. R3step sum =

=

C. Perfect and Partial HSI Processing Before proceeding to the sum-rate analysis of particular WNC relaying strategies, we provide a general discussion about the impact of HSI on WBN processing. Without loss of generality, we focus on destination DA processing (discussion for DB is similar). We denote the HSI message decoded at DA (overheard in step I from the unintended source SB transmission) as dAHSI . As we show later, dAHSI comprises only some specific part of message dB (overheard from source SB ), and hence, dAHSI itself does not carry the desired information for DA . However, dAHSI is necessary to enable the decoding of dA from the hierarchical message dR (received from the relay in step II) at the destination decoder DDA (and similarly for DDB ):
dR , dAHSI 7−→ dA , DDA : (7)
B dR , dHSI 7−→ dB . DDB : (8)

The mapping operations in (7), (8) summarize (in a simplified form) the basic principle of destination processing in all WNC strategies in WBN. DA decodes the desired data solely from dR (transmitted by the relay in step II) and dAHSI (comprised in the dB message transmitted by SB in step I), and similarly for DB . Hence, any feasible WNC strategy must guarantee that both destinations can decode their desired data from dR for an arbitrary amount of HSI (given by dAHSI , dBHSI ). In other words, a particular dR must be adapted to the actual quality of HSI channels to enable a successful decoding at both destinations (see e.g. [29]). Since HSI messages are overheard on channels with SNR γ2 (Fig. 1), the particular amount of HSI which can be decoded at DA (i.e. dAHSI ) and DB (i.e. dBHSI ) depends on the mutual relation between the source rate r and HSI channel capacity C (γ2 ). Based on this mutual relation between r and C (γ2 ), it is convenient to classify the processing in WBN system into perfect and partial HSI cases. 1) Perfect HSI processing (C (γ2 ) ≥ r): A complete source messages dA , dB can be decoded as HSI at respective unintended destinations, if the maximal source transmission rates (r) are restricted to be below the HSI channel capacity (C (γ2 )): dAHSI per f = dB , dBHSI per f = dA . 7 In the 3-step DF scheme both sources equally share the available time resources in an orthogonal way and hence each source transmits 12 NI symbols.

4

In this case, each HSI channel reliably conveys the message from unintended source to the respective destination and the WBN system becomes virtually equivalent to 2-WRC. Hence, similarly as in 2-WRC, a maximal “compression” (see e.g. [14]) of the relay hierarchical message is possible (|dR | = max {|dA | , |dB |}). Unfortunately, the performance of “perfect HSI” processing deteriorates rapidly with a decreasing capacity of the HSI channel. As C (γ2 ) → 0, the HSI channel inevitably becomes the bottleneck of the system, leading into a zero sum-rate (5), (6) for C (γ2 ) = 0. 2) Partial HSI processing (C (γ2 ) < r): From an information-theoretic point of view it becomes possible to decode only a specific part of information from the observed signal, if an optimized transmission scheme (see e.g. [30, Section IV.C.]) is employed at transmitting nodes. In this sense we can assume that DA can successfully decode some specific part of dB as its HSI message dAHSI (and similarly for DB ): 0 ≤ dAHSI ≤ C (γ2 ) , 0 ≤ dBHSI ≤ C (γ2 ) . Since the relay hierarchical message must properly respect the actual amount of partial HSI at DA , DB to guarantee the decodability of desired data at both destinations, it follows that max {|dA | , |dB |} < |dR | ≤ |dA | + |dB | .

(9)

Note that the lower bound on |dR | (strict inequality) corresponds to the perfect HSI processing, while the upper bound is equivalent to the conventional routing-based transmission scheme, which is a special case of the partial HSI processing for C (γ2 ) = diHSI ≅ 0, i ∈ {A, B} and |dR | = |dA | + |dB |. As we show in the following sections, the partial HSI processing is eligible to provide a better sum-rate performance than perfect HSI processing in some specific SNR regions, allowing to keep the source transmission rates above the HSI channel capacity (r > C (γ2 )). III. R ELAYING S TRATEGIES IN WBN

WITH

L IMITED HSI

As we have already mentioned, WBN can be always forced to operate in the perfect HSI regime, if the source rates are kept bounded below the HSI channel capacity (i.e. if r ≤ C (γ2 )). However, it can be shown that such approach is usually only sub-optimal (especially in the low γ2 region) and superior sumrate performance can be achieved if r > C (γ2 ) and partial HSI processing is allowed. The particular node operations in both perfect and partial HSI processing are summarized in Table I for DF and in Table II for JDF and HDF relaying schemes8 . Note that for C (γ2 ) → 0 the HSI channels become completely unreliable and hence separate individual source messages must be decoded and broadcast by the relay (in all DF, JDF and HDF strategies), making the WBN processing equivalent to the conventional routing approach (see Tables I, II for C (γ2 ) = 0). In the following sections we derive the maximal sum-rates rates of particular relaying strategies in WBN. We assume 8 More details about the particular processing in the AF relaying scheme will be provided later.

(without loss of generality) that the total length of step I is always kept fixed to NI symbols in all the relaying strategies, while step II is allowed to have variable length9. In our analysis we also assume that the relay is aware about the amount of HSI decoded by destinations after step I. Although this could be viewed as a relatively strong assumption, we show that some coding strategies (e.g. Superposition Coding [29]) can naturally guarantee this. To further simplify (and clarify) the analysis, we assume that each destination performs a hard decision on HSI prior to the processing of relay signal. Even though that such processing could be sub-optimal, it still allows a fair comparison of WNC relaying strategies in the limited-HSI network. Similarly as in [14] we also do not optimize the relay broadcast strategy. A. 3-step Decode & Forward Scheme Sources SA , SB equally share the available time in step I of the 3-step scheme, resulting in two orthogonal source-relay channels (SA → R, SB → R). Source signals do not mutually interfere and hence the relay is able to decode both individual source massages if a suitable transmission rate r is set at both sources. The relaying strategy where such separate decoding of orthogonal source transmissions is performed is usually called DF [14]. The individual source messages dA , dB are always decoded by the relay (from orthogonal observations), regardless of the amount of HSI which can be gathered at destinations. Since we assume that the relay knows the amount (and structure) of HSI received at destinations, it is always able to compose the output message dR from the decoded messages and then broadcast it to DA , DB (see Table I). The maximal sum-rate of the DF scheme can be evaluated as shown in the following Theorem: Theorem 2. (DF sum-rate): The maximal sum-rate in WBN with DF relaying strategy is:   C(γ1 )C(γ3 ) , γ2 ≥ γ1 C(γ3 )+0.5C( γ1 ) i h (10) = RDF 2 2′ sum DF DF max Rsum ; Rsum , γ2 < γ1 2

where RDF sum =

C(γ2 )C(γ3 ) C(γ3 )+0.5C(γ2 )

2′

and RDF sum =

C(γ1 )C(γ3 ) C(γ3 )+C(γ1 )−0.5C(γ2 ) .

Proof: The proof is available in Appendix A. Since perfect HSI can be received by both destinations whenever γ2 ≥ γ1 , WBN with DF strategy becomes equivalent to the 2-WRC scenario in this SNR region. Here the sourcerelay channels are the main bottleneck of the system and hence RDF sum does not depend on γ2 . Much more interesting situation occurs if γ2 < γ1 , i.e. if the HSI channels are the bottleneck of the system. A straightforward approach is to reduce the source rate to rDF = C (γ2 ) to guarantee a decodability of perfect HSI at destinations. Another option is to exploit the partial HSI processing (Table I), while keeping the source rate at the maximum possible value for the given relaying strategy (rDF = C (γ1 ) guarantees successful DF relay decoding). It can be easily shown that the latter approach (i.e. the partial 9 Note that in AF the relay only retransmits each received symbol [32] and hence both steps have always equal length, i.e. NI = NII .

5

Available HSI

Perfect/full

SA ,SB broadcast (step I)

Sources SA , SB orthogonally share the available time in step I and hence they successively broadcast their messages dA ,dB (|dA | = |dB | = 12 NI r) towards the relay and unintended destinations DB ,DA with rate r.

HSI channel capacity vs. source rate

C (γ2 ) ≥ r

C (γ2 ) < r

DA : HSI processing

Decodes perfectly the information message transmitted by the unintended source SB , dAHSI = 12 NI r.

(A1) Decodes only the partial HSI message, dHSI = 21 NI C (γ2 ).

DB : HSI processing

DF relay decoding

Imperfect/partial

Decodes perfectly the information message transmitted by the unintended source SA , dBHSI = 12 NI r. Decodes separate source data dA ,dB (|dA | = |dB | = 12 NI r) and combines them to form |dAB | = 12 NI r (e.g. bit-wise XOR of the messages) [17].

(B1) Decodes only the partial HSI message, dHSI = 21 NI C (γ2 ).

Decodes separate source data dA ,dB (|dA | = |dB | = 12 NI r) and then splits each data message di (i ∈ {A,B}) into (2) (1) two separate parts, where di = 21 NI C (γ2 ) and di = (1) (1) (1) |di |− di = 12 NI (r −C (γ2 )). Subsequently dA and dB are (1)

combined to form the hierarchical message dAB . i h (1) (2) (2) Forms the output message as dR = dAB ,dA ,dB , |dR | =

Relay broadcast (step II)

Broadcasts (after a potential re-encoding) the hierarchical message dR = dAB , |dR | = 12 NI r with rate rR = C (γ3 ) to both destinations.

DA decoding

Full HSI dAHSI = dB is combined with dAB to decode dA .

Partial HSI dHSI = dB is combined with dAB to obtain dA (2) and joined with dA to get dA .

DB decoding

Full HSI dBHSI = dA is combined with dAB to decode dB .

Partial HSI dHSI = dA is combined with dAB to obtain dB (2) and joined with dB to get dB .

1 2 NI C (γ2 )+NI (r −C (γ2 )) and broadcasts it (after a potential re-encoding) with rate rR = C (γ3 ) to both destinations. (A1)

(1)

(1)

(1)

(B1)

(1)

(1)

(1)

Table I N ODE OPERATIONS IN DF

RELAYING SCHEME IN PERFECT AND PARTIAL

HSI CASES .

Available HSI

Perfect/full

SA ,SB broadcast (step I)

Sources SA , SB simultaneously broadcast their messages dA ,dB (|dA | = |dB | = NI r) towards the relay and unintended destinations DB ,DA with rate r.

HSI channel capacity vs. source rate

C (γ2 ) ≥ r

C (γ2 ) < r

DA : HSI processing

Decodes perfectly the information message transmitted by the unintended source SB , dAHSI = NI r.

(A1) Decodes only the partial HSI message, dHSI = NI C (γ2 ).

JDF relay decoding

Decodes separate source data dA ,dB (|dA | = |dB | = NI r) and combines them to form |dAB | = NI r (e.g. bit-wise XOR of the messages) [17].

Decodes separate source data dA ,dB (|dA | = |dB | = NI r) and then splits each data message di (i ∈ {A,B}) into two separate (1) (2) (1) parts, where di = NI C (γ2 ) and di = |di | − di =

HDF relay decoding

Decodes directly the hierarchical data dAB (|dAB | = NI r).

Relay broadcast (step II)

Broadcasts (after a potential re-encoding) the hierarchical message dR = dAB , |dR | = NI r with rate rR = C (γ3 ) to both destinations.

DA decoding

Full HSI dAHSI = dB is combined with dAB to decode dA .

Partial HSI dHSI = dB is combined with dAB to obtain dA (2) and joined with dA to get dA .

DB decoding

Full HSI dBHSI = dA is combined with dAB to decode dB .

Partial HSI dHSI = dA is combined with dAB to obtain dB (2) and joined with dB to get dB .

DB : HSI processing

Decodes perfectly the information message transmitted by the unintended source SA , dBHSI = NI r.

Imperfect/partial

(B1) Decodes only the partial HSI message, dHSI = NI C (γ2 ). (1)

(1)

NI (r −C (γ2 )). Subsequently dA and dB are combined to (1) form the hierarchical message dAB . (1) (1) Decodes directly the partial hierarchical data dAB ( dAB = NI C (γ2 )) and separately the two remaining parts of indi (2) (2) vidual source messages di (i ∈ {A,B}), where di = (1) |di | − di = NI (r −C (γ2 )). i h (1) (2) (2) Forms the output message as dR = dAB ,dA ,dB , |dR | = NI C (γ2 )+ 2NI (r −C (γ2 )) and broadcasts it (after a potential re-encoding) with rate rR = C (γ3 ) to both destinations. (A1)

(1)

(1)

(1)

(B1)

(1)

(1)

(1)

Table II N ODE OPERATIONS IN JDF AND HDF RELAYING SCHEMES IN PERFECT AND PARTIAL HSI CASES .

6 2′

2

H

D

F

re gi

on

DF HSI processing) provides a higher (RiDF sum > Rsum ) iff h sum-rate 2 2′ 1 DF DF C (γ3 ) > 2 C (γ2 ) and hence max Rsum ; Rsum in (10) can be further simplified according to this mutual relation between C (γ3 ) and C (γ2 ).

Theorem 3. (AF sum-rate): The maximal sum-rate in WBN with AF relaying strategy is:     γ2 ≥ 2γ1γ+1 γγ33 +1 C 2γ1γ+1 γγ33 +1 ,   γ1 γ3 γ1 γ3 γ ), RAF sum = C ( 2γ1 +γ1 γ3 +γ3 +1 ≤ γ2 < 2γ1 +γ3 +1  2   γ1 γ3 γ1 γ3 C , γ < 2γ1 +γ1 γ3 +γ3 +1

2

2γ1 +γ1 γ3 +γ3 +1

(11)

Proof: The proof is available in Appendix B. 2) Joint Decode & Forward: The JDF relay always decodes the individual source messages dA , dB from MAC channel observation (see Fig. 1), regardless of the amount of HSI which can be gathered at destinations. Since we assume that the relay knows the amount (and structure) of HSI received at destinations, it is always able to compose the output message dR and then broadcast it to DA , DB (see Table II). The maximal sum-rate of the JDF scheme can be evaluated as shown in the following Theorem: Theorem 4. (JDF sum-rate): The maximal sum-rate in WBN

F JD

Both sources SA , SB are allowed to transmit simultaneously in step I in all the 2-step schemes and hence the relay has only a single compound observation of source signals in the Multiple-Access Channel (MAC) (Fig. 1). Several 2-step relaying strategies can be distinguished in WBN according to a particular processing of this compound signal, including AF, JDF and HDF. 1) Amplify & Forward: The AF relay always only amplifies the received signal prior to broadcasting it to both destinations [14], [32]. Since the AF relay does not decode the source messages from the received signal, it is not able to modify its output message (contrary to DF, JDF, HDF – see Tables I,II) to respect the actual HSI observed at destinations. Each destination thus receives a mixture of source signals from the relay, having its desired data interfered by the data from the unintended source. This creates two equivalent channels SA → DA , SB → DB , where the particular equivalent SNR depends mainly on the destinations’ ability to remove the interfering signal. The interfering signal can be perfectly removed iff the destination is able to decode the unintended source message overheard on HSI channel, i.e. iff r ≤ C (γ2 ) . On the other hand, if the destination is not able to decode the HSI signal (i.e. if r > C (γ2 )), it can either estimate its value from the HSI channel observation (and subtract this estimate from the received relay signal) or completely ignore the HSI channel observation and treat the interfering signal as an additional noise. The maximal rate of AF strategy is proved in the following Theorem:

re gi on

B. 2-step Schemes

Figure 2. Operational regions of HDF strategy: impact of the relationship between the HSI channel capacity C (γ2 ) and relay MAC capacity region.

with JDF relaying strategy is:   C(2γ1 )C(γ3 ) , C (γ2 ) ≥ 21 C (2γ1 ) C(γ3 )+0.5C(2 γ1 ) i h ′ RJDF = 2 sum JDF 2 , C (γ ) < 1 C (2γ ) max RJDF 2 1 sum ; Rsum 2 2

where RJDF sum =

2C(γ2 )C(γ3 ) C(γ3 )+C(γ2 )

2′

and RJDF sum =

(12)

C(2γ1 )C(γ3 ) . C(γ3 )+C(2γ1 )−C(γ2 )

Proof: The proof is available in Appendix C. It can be easily shown that for C (γ2 ) < 12 C (2γ1 ) the partial HSI processing in JDF provides a better sum-rate than the ′ JDF 2 JDF 2 perfect h one2 (Rsum2′ i > Rsum ) iff C (γ3 ) > C (γ2 ) and hence JDF in (12) can be further simplified accordmax RJDF sum ; Rsum ing to this mutual relation between C (γ2 ) and C (γ3 ). 3) Hierarchical Decode & Forward: The fundamental idea of HDF relaying strategies (see [16], [17], [20], [31] and references therein) is based on the fact that the intermediate relay node is not the final destination of communication and hence it does not have to decode separate individual source messages. This in turn allows to increase the source rates above the limits of the underlying MAC capacity region [33, Chapter 15] (compare with JDF) at the relay, and hence to further boost the sum-rate performance. However, a sufficient amount of HSI must be available at destinations to gather this potential performance benefit of HDF. In other words, HDF provides higher sum-rates than JDF only if HSI channels support the rates above the MAC capacity region (i.e. if C (γ2 ) ≥ 12 C (2γ1 )) This is illustrated in Fig. 2. When sufficient HSI is available, the HDF relay decodes directly the hierarchical message dR = dAB from its observations10 , and hence the individual source messages dA , dB are not necessarily decoded. Since we assume that the relay knows the amount (and structure) of HSI received at destinations, it is always able to compose the output message dR and then broadcast it to DA , DB (see Table II). The maximal sum-rate of the HDF scheme can be evaluated as shown in the following Theorem: 10 d AB is generally some invertible function of source data. Each destination can decode its desired data from dAB iff data from the unintended source (HSI) are also available [31].

7

6

6 γ =γ

γ =γ

1

2

5

5

4

4 Two−way rate [bps/Hz]

Two−way rate [bps/Hz]

2

3

2

3

2 C(γ ) = 0.5C(2γ ) 2

C(γ ) = 0.5C(2γ )

1

2

1

−15

−10

−5

0

5 10 SNR γ2 [dB]

15

20

25

30

Theorem 5. (HDF sum-rate): The maximal sum-rate in WBN with HDF relaying strategy is:  2C(γ )C(γ ) 1 3  , C (γ2 ) ≥ C (γ1 )   C(γ1 )+C(γ3 ) 2C(γ2 )C(γ3 ) 1 HDF Rsum = C(γ2 )+C(γ3 ) , 2 C (2γ1 ) ≤ C (γ2 ) < C (γ1 ) i h  ′  max RHDF 3 ; RHDF 3 , C (γ ) < 1 C (2γ ) 3

where RHDF sum =

2

sum

2C(γ2 )C(γ3 ) C(γ3 )+C(γ2 )

3′

= and RHDF sum

2

1

C(2γ1 )C(γ3 ) C(γ3 )+C(2γ1 )−C(γ2

(13) ).

Proof: The proof is available in Appendix D.

3−step DF 2−step AF 2−step JDF 2−step HDF/DNF

0 −20

35

Figure 3. Comparison of the maximal sum-rates of DF, AF, JDF and HDF/DNF strategies in WBN (γ1 = 10 dB, γ3 = 30 dB).

sum

1

1

3−step DF 2−step AF 2−step JDF 2−step HDF/DNF

−15

−10

−5

0

5 10 SNR γ2 [dB]

15

20

25

30

35

Figure 4. Comparison of the maximal sum-rates of DF, AF, JDF and HDF/DNF strategies in WBN (γ1 = 30 dB, γ3 = 10 dB). 10 γ2 = γ1 9

8

7 Two−way rate [bps/Hz]

0 −20

1

6

5

4

3

C. Performance Comparison In this section we finally compare the performance of relaying schemes in WBN. Besides of the sum-rates of particular strategies we evaluate the corresponding relative lengths of step II in all relaying strategies to emphasize the fact that WBN must operate with uneven lengths of steps I, II to achieve the sum-rates evaluated in the previous section. Since we are interested mainly in the impact of limited HSI in this paper, the sum-rates and relative lengths of step II of particular relaying strategies are compared as a function of the HSI channel SNR (γ2 ). 1) Maximal Sum-Rates: Maximal sum-rates of the analysed relaying schemes are compared in Fig. 3 (γ1 = 10dB, γ3 = 30dB), Fig. 4 (γ1 = 30dB, γ3 = 10dB) and Fig. 5 (γ1 = γ3 = 30dB). The values of γ2 where C (γ2 ) is equal to the 1st and 2nd order (symmetric rates) cut-set bounds of the underlying relay MAC channel are emphasized in all Figures. These cutset bounds determine the HSI operating regions in most of the strategies (see equations (10), (12), (13)). Many interesting observations can be made from Figs. 3, 4, 5. Similarly as in the 2-WRC scenario [14], the best sumrate is provided by the HDF/DNF strategy in the whole range of channel SNRs11 . However, when C (γ2 ) ≤ 21 C (2γ1 ) the 11 Note that in the low to medium γ region only a lower bound on AF 2 strategy sum-rate is provided (see proof of Theorem 3).

C(γ ) = 0.5C(2γ ) 2

1

2 3−step DF 2−step AF 2−step JDF 2−step HDF/DNF

1

0 −20

−15

−10

−5

0

5 10 SNR γ2 [dB]

15

20

25

30

35

Figure 5. Comparison of the maximal sum-rates of DF, AF, JDF and HDF/DNF strategies in WBN (γ1 = 30 dB, γ3 = 30 dB).

performance of HDF degrades theoretically to that of the JDF strategy, since the actual quality of HSI channels does not allow to increase the source rates above the conventional MAC capacity region (see Fig. 2). The sum-rates depend significantly on the actual SNR of HSI channels (γ2 ) and all the relaying strategies are capable to support non-zero sum-rates even when the HSI channels are very weak (C (γ2 ) → 0). The impact of HSI channels on the maximal sum-rate becomes negligible in both low (γ2 → 0) and high (γ2 > γ1 ) γ2 regions (for fixed γ1 , γ3 ), where the maximal sum-rates of all WNC strategies tend to saturate. The particular regions of γ2 where this saturation of sum-rates can be observed depend mainly on the applied relaying strategy and also on the actual values of γ1 , γ3 . 2) Relative Length of Communication Steps: The WBN system must operate with uneven lengths of step I, II to achieve

8

1

1 γ =γ

γ =γ

1

2

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6 n [−]

0.5

II

nII [−]

2

0.4

1

0.5

0.4 C(γ ) = 0.5C(2γ ) 2

0.3

1

0.3 C(γ2) = 0.5C(2γ1)

0.2

0.2 3−step DF 2−step AF 2−step JDF 2−step DNF/HDF

0.1

0 −20

−15

−10

−5

0

5 10 SNR γ [dB]

15

20

25

30

35

2

1 γ =γ 2

1

0.9

0.8

0.7

nII [−]

0.6

0.5

0.4

0.3 C(γ2) = 0.5C(2γ1) 0.2 3−step DF 2−step AF 2−step JDF 2−step DNF/HDF

0.1

−15

−10

−5

0

5 10 SNR γ2 [dB]

0 −20

−15

−10

−5

0

5 10 SNR γ [dB]

15

20

25

30

35

2

Figure 6. Comparison of the relative length of step II in DF, AF, JDF and HDF/DNF strategies in WBN (γ1 = 10 dB, γ3 = 30 dB).

0 −20

3−step DF 2−step AF 2−step JDF 2−step DNF/HDF

0.1

15

20

25

30

35

Figure 7. Comparison of the relative length of step II in DF, AF, JDF and HDF/DNF strategies in WBN (γ1 = 30 dB, γ3 = 10 dB).

the sum-rates evaluated in (10), (12), (13). Since the relay itself is not a source of information (it does not have any own data to transmit) the length of step II is always proportional to the length of step I, i.e. NII = δ NI (note again that NI is kept fixed in all strategies), where δ is the proportional coefficient12. The sum-rate in (5) can be evaluated as:

Figure 8. Comparison of the relative length of step II in DF, AF, JDF and HDF/DNF strategies in WBN (γ1 = 30 dB, γ3 = 30 dB).

From (14), (15) is is obvious that the corresponding optimal relative lengths of step II (nII = 1 − nI ) can be directly obtained from the sum-rates (as proved in Theorems 2, 4, 5) and hence we omit a detailed derivation of nII in this paper. The optimal values of nII which correspond to the sumrates in Figs. 3, 4, 5 are compared in Fig. 6 (γ1 = 10dB, γ3 = 30dB), Fig. 7 (γ1 = 30dB, γ3 = 10dB) and Fig. 8 (γ1 = γ3 = 30dB). The lengths of step I, II must be always optimized to achieve the optimal sum-rate performance in WBN. For example, less resources are required in step II when γ1 < γ3 , due to the superior quality of relay→destination channels (see Fig. 6). On the other hand, the length of step II must be increased relatively to step I when γ1 > γ3 , i.e. when the relay→destination channels are weak (see Fig. 7). The particular values of γ2 where nII curves change its derivative (in DF, JDF and HDF) in Fig. 7 correspond to the value of γ2 where the partial HSI processing becomes superior than the perfect HSI processing. As noted in the previous discussion this corresponds to C (γ3 ) = 21 C (γ2 ) in DF strategy and to C (γ3 ) = C (γ2 ) in JDF and HDF strategies.

(15)

IV. S UPERPOSITION C ODING BASED HDF S CHEME So far we have assumed that the employed coding strategy at SA , SB guarantees that the relay node is aware about the amount of HSI received at destinations13, and moreover, we have focused on a hypothetical optimized coding strategy (see e.g. [30, Section IV.C.]), that allows to decode the partial HSI messages at the maximal possible rate ( dAHSI = dBHSI = C (γ2 )). In the rest of this paper we show that Superposition Coding (SC) [29] is a feasible coding strategy for relaying in WBN. We introduce the SC-based HDF14 relaying scheme (HDFSC ), which is eligible to guarantee a fulfilment of the first assumption (relay is aware about the structure of HSI at destinations).

steps have a constant length (NI = NII , δAF = 1) in the AF strategy where the relay only retransmits each received symbol [32].

13 Note again that this assumption secures that the relay is able to construct its output message for arbitrary HSI at destinations. 14 SC can be similarly implemented also in the DF, AF and JDF strategies.

R2step sum =

2r = 2rnI , 1+δ

(14)

NI where nI = NI +N = 1+1 δ is the relative length of step I, and II similarly for the 3-step strategy (6):

R3step sum = 12 Both

r = rnI . 1+δ

9

Lemma 6. (HDFSC sum-rate): In step I sources SA , SB transmit simultaneously their messages of NI symbols with rate r = rb + rs . The maximal sum-rate rate of HDFSC in WBN is then given by: HSI

HSI

SC (γ , γ , γ ) ≤ RHDF 1 2 3 sum




2C (γ3 ) rb αopt + rs αopt

, C (γ3 ) + rb αopt + 2rs αopt

(17)

where αopt is the optimal value of SC power division parameter maximizing
the sum-rate (17) for a
given SNR triplet γ1 , γ2 , γ3 , and rb αopt (respectively rs αopt ) is the optimized rate of the basic (respectively superposed) message. MAC phase

BC phase

Figure 9. Principle of HDFSC processing in WBN. Symbol ⊕ denotes a hierarchical function (e.g. bit-wise XOR).

Proof: Total of NI (rA + rB ) = 2NI r = 2NI (rb + rs ) bits are transmitted simultaneously from sources SA , SB in step I. The relay must send separately three independent messages (16) in step II through the relay→destination channel with capacity rb NI 2rs NI C (γ3 ), and hence step II will be at least NII = C( γ ) + C(γ ) 3

We provide a numerical optimization of HDFSC parameters, and we compare the maximal sum-rate of HDFSC with the maximal sum-rate RHDF sum of the hypothetical optimized coding strategy. A. Implementation of SC-based Relaying Scheme in WBN Basic principle of HDFSC processing is visualized in Fig. 9 (see [29] for more details). Both sources Si , i ∈ {A, B} split their messages into two independent (basic & superposed) parts of potentially unequal length and broadcast them simultaneously in step I. The m-th complex baseband √ transmitted symbol from source Si is hence xi [m] = 1 − αixbi [m] + √ s αi xi [m], where 0 ≤ αi ≤ 1 is the SC power division parameter. By superscript b (respectively s) we denote the basic (respectively superposed) message. The fundamental idea of HDFSC is to process only the basic messages as an effective HSI at destinations, treating the superposed messages on overheard source-destination channels purely and simply as an additional interference (with obvious consequences on the HSI channels capacity) [29]. Since the basic messages are decoded at both unintended destinations as HSI, it is sufficient to decode a hierarchical function of basic messages (e.g. bit-wise XOR [17]) at the relay, while both individual superposed messages must be decoded separately. The relay then successively transmits (after a potential reencoding) the hierarchical basic message and two individual superposed messages, forming a relay output message as h i dR = dbA ⊕ dbB , dsA , dsB , (16) where ⊕ denotes a hierarchical function (e.g. bit-wise XOR [17]). Each destination  Di ,i ∈ {A, B} is able to decode the desired message di = dbi , dsi iff both the relay message dR and effective HSI (given by the basic message dbj , j ∈ {A, B} , j 6= i) are correctly decoded (Fig. 9). As shown in the following Lemma, it is also straightforward to evaluate the maximal sum-rate of the HDFSC scheme as a function of the rates of superposed and basic messages.

HDF

3

symbols long. Consequently, from (5) we have Rsum SC ≤ 2NI (rb +rs ) NI +NII , which gives us finally (17). The SC power division parameter α becomes obviously the crucial element of the HDFSC system design. As shown in [29] a numerical optimization of α must be performed to find the maximum sum-rate (17) of the SC-based strategy. We perform this optimization for HDFSC and compare its maximal sumrate with RHDF sum (evaluated in Theorem 5). B. HDFSC relaying strategy To provide a fair comparison with RHDF sum (see Theorem 5) we must follow the assumption of independent processing of HSI and relay channel observations (as stated in Section II-B). Thus, we assume that in the HDFSC strategy each destination Di , i ∈ {A, B} performs a decision on HSI (basic message dbj , j ∈ {A, B} , j 6= i) immediately after step I, i.e. prior to the reception of the relay signal. A numerical optimization of α must be performed to find the rates rb , rs maximizing the sum-rate in (17). This optimization was performed already in [29] for a special case of relay processing which assumes a specific interference cancellation of the basic messages at the relay. However, since a feasibility of such processing has not yet been proven, in this paper we modify the relay processing in the HDFSC strategy according to the results of the analysis in Section III. When C (γ2 ) ≥ 12 C (2γ1 ) we assume that the rates of superposed messages are set to zero (rs = 0) and only the basic messages are transmitted. Since the source rate r = rb = min [C (γ2 ) , C (γ1 )] is kept bounded below the HSI channel capacity in this region, a perfect HSI processing (see Table II) is feasible. On the other hand (as noted in Section III) we assume that the HDF operation is equivalent to JDF when C (γ2 ) < 12 C (2γ1 ). Here, the relay decodes separately both basic and also both superposed messages, which corresponds to an equivalent 4-user MAC decoding15. This, together with the assumption of destination’s HSI decoding after step I (as mentioned above), provides a set of rate upper-bounds 15 Note that hierarchical message db ⊕ db can be always constructed when B A both separate basic messages are decoded by the relay.

10

for basic and superposed messages, as summarized in the following Lemma:

10 rb 9

r=r +r

Lemma 7. (HDFSC messages rates): If C (γ2 ) < 21 C (2γ1 ) the HDFSC relay operates in the JDF mode, and hence the rates of basic (rb ) and superposed messages (rs ) are bounded by the following cut-set bounds:

2rs (α ) ≤ C (2αγ1 ) , 2rb (α ) ≤ C (2 (1 − α ) γ1 ) ,

rb (α ) + rs (α ) ≤ C (γ1 ) , 2rs (α ) + rb (α ) ≤ C ((1 + α ) γ1 ) ,

rs (α ) + 2rb (α ) ≤ C ((2 − α ) γ1 ) , 2 (rb (α ) + rs (α )) ≤ C (2γ1 ) .

s

C(γ ) 2

7

Rate [bps/Hz]

(18) (19)

b

8

6

5

4

(20) (21)

3 C(γ2) = 0.5C(2γ1)

(22) (23)

2

(24) (25)

The HSI (basic message) is decoded at destinations after step I from the interference channel, bounding the rate of basic messages as:   (1 − α ) γ2 rb (α ) ≤ C . (26) 1 + αγ2 Proof: Due to the symmetry of the analysed WBN we can assume without loss of generality that rA = rB = r = rb + rs and hence αA = αB . In step I the relay has the following observation:  √ √ yR =hSA R 1 − α xbA + α xsA +  √ √ (27) 1 − α xbB + α xsB + zR h SB R

which forms an equivalent 4-user MAC channel. Therefore, a set of upper-bounds of achievable rates for basic and superposed messages can be directly evaluated as a set of particular cut-set bounds [33, Chapter 15] of this channel. We can immediately evaluate the 1st order (18), (19), 2nd order (20), (21), (22), 3rd order (23), (24) and 4th order (25) cut-set bounds. The last bound in Lemma 7 (26) corresponds to the decoding of √ effective HSI
the interference HSI √ from channel yS j Di = h ji 1 − α xbi + α xsi + zDi at destination Di , i, j ∈ {A, B} and i 6= j. Based on the rate bounds evaluated in Lemma 7 a numerical optimization of α can be performed (see
[29] for
more details) to provide the optimal rates rb αopt , rs αopt which maximize the HDFSC sum-rate (17). An example of particular optimized values of α , rb , rs is evaluated for γ1 = γ3 = 30 dB in Fig. 10 as a function of γ2 (compare the results with the proof of Theorem 5). A comparison of the maximal sum-rate of HDFSC with the maximal sum-rate of HDF strategy RHDF sum (13) is available in Fig. 11. As expected, when C (γ2 ) ≥ 12 C (2γ1 ), the HDFSC strategy can provide the same sum-rate as RHDF sum . The rates of superposed messages are set to zero (see Fig. 10) in this region, and hence a perfect HSI processing can be always performed (see Table II). On the other hand, if C (γ2 ) < 21 C (2γ1 ) the HDFSC strategy does not always achieve the maximal sum-rate RHDF sum in the whole range of observed channel SNRs γ1 , γ2 , γ3 (see

α

1

0 −20

opt

−15

[−]

−10

−5

0

5 10 SNR γ [dB]

15

20

25

30

35

2

Figure 10. Optimized values of α , rb , rs as a function of γ2 (γ1 = γ3 = 30 dB). Note that for C (γ2 ) ≥ 21 C (2γ1 ) the source rate is set to r = rb = min {C (γ2 ) ,C (γ1 )} and hence the system operates with a perfect HSI, while for C (γ2 ) < 12 C (2γ1 ) the source rate r ≥ C (γ2 ) and consequently the system operates with a partial HSI. 10 HDF

Rsum ; γ1=30 dB, γ3=10 dB RHDF−SC; γ =10 dB, γ =30 dB 9

sum HDF

1

3

Rsum ; γ1=10 dB, γ3=30 dB RHDF−SC ; γ1=10 dB, γ3=30 dB sum

8 Two−way rate [bps/Hz]

rs (α ) ≤ C (αγ1 ) , rb (α ) ≤ C ((1 − α ) γ1 ) ,

γ2 = γ1

rs

RHDF; γ =γ =30 dB sum

1

3

RHDF−SC ; γ1=γ3=30 dB sum 7

6

5

4

3 −15

−10

−5

0

5

10 SNR γ2 [dB]

15

20

25

30

35

Figure 11. Comparison of the upper-bound of the HDFSC sum-rates with RHDF sum .

Fig. 11). A similar information-theoretic sub-optimality of SCprocessing was already observed in [30] for the conventional one-way relay channel. V. C ONCLUSION In this paper we have shown that all contemporary bidirectional WNC strategies can be appropriately modified to provide a non-zero rate in WBN with generally limited (partial) HSI. If the HSI channels are not the bottleneck of the system (i.e. if C (γ2 ) ≥ C (γ1 )), WBN becomes equivalent to 2-WRC and hence the perfect HSI processing is feasible. Much more interesting situation occurs when γ2 < γ1 , i.e. when the HSI channel capacities potentially limit the system

11

performance. Generally, there are two ways how to deal with this problem. A straightforward solution is to restrict the source rate below the HSI channel capacity (rmax ≤ C (γ2 )) to maintain the availability of perfect HSI at destinations. However, at some point in the SNR region (depending also on the particular WNC strategy) this approach becomes suboptimal. We have shown that a modified WNC processing, exploiting only a specific part of source information as the effective (partial) HSI, is eligible to provide a superior sumrate performance. The presented information-theoretic analysis has revealed that some sort of feedback in WBN is necessary to make a suitable choice of the relay processing strategy (perfect/partial HSI) and to adapt the transmission rates to the actual channel conditions. We have shown that SC-based HDF strategy (HDFSC ) provides a natural information-theoretic tool for the implementation of adaptive HDF relaying in WBN. The HDFSC strategy can be considered as a first step towards a practical implementation of adaptive modulation-coding scheme (as shown e.g. in [30, Section V.]), allowing to adapt the WBN processing to arbitrary quality of HSI channels by a proper distribution of power among the basic and superposed messages of the underlying SC scheme.

P ROOF

OF

A PPENDIX A T HEOREM 2 (DF S UM - RATE )

Step I is time-shared by both sources and hence the sources transmit separately their messages of 12 NI symbols to the relay. The optimal value of source rate rDF which maximizes the sum-rate depends on the mutual relation between γ1 and γ2 and hence we split the proof into two disjunct γ2 regions: •

1

DF = C (γ ) guarantees that both (γ2 ≥ γ1 ): Source rate rmax 1 individual source messages can be decoded by the relay. DF 1 = C (γ ) in this region, source mesSince C (γ2 ) ≥ rmax 1 sages can be decoded also by the unintended destinations in step I and hence the perfect HSI processing can be applied according to Table I. After decoding of source messages, the relay broadcasts the output message of size |dR | = |dAB | = 21 NI C (γ1 ) with rate rR = C (γ3 ) to ensure a decodability of this message at both destinations and hence the sum-rate (6) can be evaluated as: 1

RDF sum =

1 1 2 NI C (γ1 ) + 2 NI C (γ1 )

NI + NII

=

N C (γ1 )  I . 1 C(γ ) 1 2 NI 1 + C(γ ) 3

•

(A.1) (γ2 < γ1 ): The HSI channels are the bottleneck of the system in this region. There are two possible solutions how to cope with this problem. If the source rate is DF 2 decreased to rmax = C (γ2 ) we can still guarantee that perfect HSI is retrieved by destinations, which leads to the following sum-rate (similarly as in (A.1)): 2

RDF sum =

1 1 2 NI C (γ2 ) + 2 NI C (γ2 )

NI + NII

=

N C (γ2 )  I . 1 C(γ ) NI 1 + 2C(γ 2) 3

(A.2)

2′

DF = The second option is to keep the source rate at rmax 1 DF rmax = C (γ1 ), at the price of having only partial HSI. ′ DF 2 = C (γ ) in this region, only partial Since C (γ2 ) < rmax 1 HSI can be received by destinations and the partial HSI processing must be applied according to Table I. After decoding of source hmessages, the irelay broadcasts (1) (2) (2) the output message dR = dAB , dA , dB of size |dR | =
1 1 2 NI C (γ2 ) + 2 2 NI (C (γ1 ) − C (γ2 )) with rate rR = C (γ3 ) to ensure a decodability of this message at both destinations and hence the sum-rate (6) can be evaluated as: 2′

RDF sum =

1 1 2 NI C (γ1 ) + 2 NI C (γ1 )

NI + NII

=

NI C (γ1 ) .  C(γ )− 1 C(γ ) NI 1 + 1C(γ2 ) 2 3

(A.3) Although it is possible to explicitly evaluate γ2 , γ3 re2′ DF 2 gion where RDF sum > Rsum , for a better clarity of results hwe express ithe sum-rate for γ2 < γ1 simply as ′ 2 DF 2 max RDF sum ; Rsum . This gives us finally (10). P ROOF

OF

A PPENDIX B T HEOREM 3 (AF S UM - RATE )

In step I both sources transmit their messages of NI symbols simultaneously to the relay. The relay has the following observation: yR = hSA R xA + hSB R xB + zR . (B.1) After receiving (B.1), the relay simply multiplies it by the AF amplification factor β [14], [32]: s s 1 1 β= , (B.2) = 2 2 N0 (2γ1 + 1) hS R + |hS R | + N0 B

A

to keep the mean energy per symbol constant, and broadcasts the resulting signal xR = β yR to destinations. Without loss of generality we describe processing at DA (DB processing follows the same steps). DA receives the following signal from the relay: yRDA = hRDA xR = β hRDA hSA R xA + β hRDA hSB R xB + β hRDA zR + zDA . (B.3) The interfering signal xB can be completely removed from yRDA iff perfect HSI is received in step I, resulting in an equiperf−HSI valent interference free SA → DA channel with SNR γAF (similarly also for SB → DB ): 2 β 2 |hRDA |2 hSA R γ1 γ3 perf−HSI  = γAF . = 2 + 2 γ γ3 + 1 2 1 N β |hRD | + 1 0

A



 AF 1 = C γ perf−HSI The source rate rmax guarantees that the AF desired message can be decoded from the equivalent channel and the sum-rate (5) can be evaluated as:     perf−HSI perf−HSI   + N C γ γ N C I I AF AF 1 perf−HSI , = C γAF RAF sum = NI + NII (B.4)

12

since NII = NI in AF.  Note that (B.4) can be achieved iff perf−HSI C (γ2 ) ≥ C γAF , as perfect HSI must be available at destinations to allow a perfect removal of the interfering signal at destinations. Now we have to analyse the sum-rate in the region where  perf−HSI C (γ2 ) < C γAF , i.e. in the region where the HSI channels are the bottleneck of the system. Perfect HSI (and consequently a perfect cancellation of the interfering signal) can be guaranteed even in this region, simply by reducing AF 2 = C (γ ). Since the equivalent sourcethe source rate to rmax 2 perf−HSI destination channels still have SNR γAF = 2γ1γ+1 γγ33 +1 > γ2 in this region, the desired source message can be decoded at each destination and hence the sum-rate (5) reduces to: NI C (γ2 ) + NI C (γ2 ) (B.5) = C (γ2 ) . NI + NII   perf−HSI The other option in this region (C (γ2 ) < C γAF ) is   ′ 1 perf−HSI AF AF and to keep the source rate at rmax = rmax = C γAF employ the AF equivalent of partial HSI processing. Since ′ AF , D C (γ2 ) < rmax A cannot perfectly decode the source SB HSI message from its HSI observation yHSI SB DA = hSB DA xB + zDA , but it can always try to estimate the value of HSI (ˆxB) and use it to (at least partially) remove the interfering signal xB from (B.3). However, a particular implementation of this partialHSI processing in AF (together with a particular method for partial HSI estimation) is a standalone research problem and hence it is beyond the scope of this paper16. For the purpose of this paper we assume that AF operates with full HSI in this region, keeping the sum-rate equal to (B.5). This generally provides only  a lower bound of the AF maximal sum-rate for perf−HSI . C (γ2 ) < C γAF

P ROOF

•

2

(C (γ2 ) ≥ 12 C (2γ1 )): Maximum symmetric source rate which guarantees that both individual source messages JDF 1 = 1 C (2γ ) [33, can be decoded by the relay is rmax 1 2 1 JDF = 1 C (2γ ) in this Chapter 15]. Since C (γ2 ) ≥ rmax 1 2 region, source messages can be decoded also by the unintended destinations in step I and hence the perfect HSI processing can be applied according to Table II. After decoding of source messages, the relay broadcasts the output message of size |dR | = |dAB | = NI 12 C (2γ1 ) with rate rR = C (γ3 ) to ensure a decodability of this message at both destinations and hence the sum-rate (5) can be evaluated as: 1

RJDF sum =

NI 21 C (2γ1 ) + NI 12 C (2γ1 ) = NI + NII

N C (2γ1 ) I . 1 C(2γ ) NI 1 + 2C(γ )1 3

•

(C.1) (C (γ2 ) < 12 C (2γ1 )): The HSI channels are the bottleneck of the system in this region. There are two possible solutions how to cope with this problem. If the source JDF 2 = C (γ ) we can still guarantee rate is decreased to rmax 2 that perfect HSI is retrieved by destinations, which leads to the following sum-rate (similarly as in (C.1)): 2

RJDF sum =

AF 2

16 The optimal HSI estimator (and its performance) can generally depend also on the employed source modulation/coding strategy.

A PPENDIX C T HEOREM 4 (JDF S UM - RATE )

In step I both sources transmit their messages of NI symbols simultaneously to the relay. The optimal value of source rate rJDF which maximizes the sum-rate depends on the mutual relation of γ1 and γ2 and hence we split the proof into two disjunct γ2 regions:

RAF sum =

It is obvious that Rsum = C (γ2 ) deteriorates rapidly with the quality of HSI channels. Fortunately, since each destination observes an analogue superposition of source messages from AF relay, it can completely ignore the unreliable HSI observation and try to decode the desired message directly from the relay signal (B.3). The resulting equivalent channel zero−HSI has SNR γAF : 2 β 2 |hRDA |2 hSA R zero−HSI   γAF = β 2 |hRDA |2 |hSB R |2 + N0 β 2 |hRDA |2 + 1 γ1 γ3 . = γ1 γ3 + 2γ1 + γ3 + 1
AF 3 = C γ zero−HSI guarantees a decodability The source rate rmax AF of the desired source message from the interference channel (B.3) and hence the sum-rate (5) can be evaluated as:

zero−HSI zero−HSI
NI C γAF + NI C γAF AF 3 zero−HSI = C γAF Rsum = . NI + NII (B.6) 3 AF 2 iff Now it is straightforward to show that RAF > R sum sum zero−HSI > γ2 which gives us finally (11). γAF

OF

2NI C (γ2 ) NI C (γ2 ) + NI C (γ2 ) . =  C(γ2 ) NI + NII NI 1 + C( γ )

(C.2)

3

2′

JDF = The second option is to keep the source rate at rmax 1 1 JDF rmax = 2 C (2γ1 ), at the price of having only partial 2′

JDF = 1 C (2γ ) in this region, HSI. Since C (γ2 ) < rmax 1 2 only partial HSI can be received by destinations and the partial HSI processing must be applied according to Table II. After decoding of source messages, the relay h i (1) (2) (2) broadcasts the output message dR = dAB , dA , dB of
size |dR | = NI C (γ2 ) + 2NI 12 C (2γ1 ) − C (γ2 ) with rate rR = C (γ3 ) to ensure a decodability of this message at both destinations and hence the sum-rate (5) can be evaluated as:

NI 12 C (2γ1 ) + NI 12 C (2γ1 ) NI C (2γ1 ) . =  C(2γ1 )−C(γ2 ) NI + NII NI 1 + C(γ3 ) (C.3) Although it is possible to explicitly evaluate γ2 , γ3 region 2′ JDF 2 where RJDF sum > Rsum , for a better clarity of results we 1 express i for C (γ2 ) < 2 C (2γ1 ) simply as h the sum-rate 2′

RJDF sum =

2

JDF max RJDF sum ; Rsum

2′

. This gives us finally (12).

A PPENDIX D

13

P ROOF

OF

T HEOREM 5 (HDF S UM - RATE )

In step I both sources transmit their messages of NI symbols simultaneously to the relay. The optimal value of source rate rHDF which maximizes the sum-rate depends on the mutual relation of γ1 and γ2 and hence we split the proof into three disjunct γ2 regions: • (C (γ2 ) ≥ C (γ1 )): In the HDF strategy, the source rates are theoretically upper bounded only by the 1st order cutset bound of the underlying MAC channel [33, Chapter 15], i.e. by r = C (γ1 ). Even though the achievability of this rate has not been rigorously proved in general Gaussian channels (see e.g. [22]), we follow the conjecture from [14] and assume that hierarchical message dAB can be successfully decoded by the relay if the sources HDF 1 transmit with the rate rmax = C (γ1 ) − ε . By setting ε = 0 we obtain the upper-bound of the maximal sumHDF 1 = C (γ ) rate of the HDF strategy. Since C (γ2 ) ≥ rmax 1 in this region, source messages can be decoded by the unintended destinations in step I and hence the perfect HSI processing can be applied according to Table II. After decoding of source messages, the relay broadcasts the output message of size |dR | = |dAB | = NI C (γ1 ) with rate rR = C (γ3 ) to ensure a decodability of this message at both destinations and hence the sum-rate (5) can be evaluated as: 1

RHDF sum =

2NI C (γ1 ) NI C (γ1 ) + NI C (γ1 )  . (D.1) =  C(γ1 ) NI + NII NI 1 + C( γ ) 3

•

( 12 C (2γ1 ) ≤ C (γ2 ) < C (γ1 )): The HSI channels become the bottleneck of the system in this region. Perfect HSI can still be provided if the source rate is reduced to HDF 2 rmax = C (γ2 ), which results in the following sum-rate: 2

RHDF sum =

NI C (γ2 ) + NI C (γ2 ) 2NI C (γ2 )  . (D.2) =  C(γ2 ) NI + NII NI 1 + C( γ ) 3

•

(C (γ2 ) ≤ 12 C (2γ1 )): In this region the HSI channels cannot support the rates above the limits induced by the relay MAC capacity region and hence the HDF strategy becomes equivalent to JDF. Source rate can be set to HDF 3 = rJDF 1 = 1 C (2γ ) and hence the maximal sumrmax 1 max 2 3 rate his the same asi in the JDF strategy, i.e. RHDF sum = ′ 2 JDF 2 , which gives us finally (13). max RJDF sum ; Rsum R EFERENCES

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Tomas Uricar received the M.Sc. (with distinction) and Ph.D. degrees in electrical engineering from Czech Technical University in Prague, Prague, Czech Republic, in 2008 and 2014, respectively. He is currently a research associate at the Department of Radio Engineering, Faculty of Electrical Engineering, Czech Technical University in Prague. He has participated in a number of research projects financed by the European Union and national agencies. His research interests include wireless communication and information theory and physical/wireless network coding. Dr. Uricar has served on various IEEE conferences as Technical Program Committee member. In 2012 he was recognized as the exemplary reviewer of the IEEE Communications Letters.

Bin Qian (S’13) received the B.S. degree in electronic and information engineering from the Harbin Institute of Technology, Harbin, China, in 2011. He is currently working towards the Ph.D. degree in electronic and computer engineering at the Hong Kong University of Science and Technology. His research interests include channel coding, wireless cooperative communications and physical layer network coding.

Jan Sykora (M’03) received the M.Sc. and Ph.D. degrees in electrical engineering from Czech Technical University in Prague, Prague, Czech Republic, in 1987 and 1993, respectively. Since 1991, he has been with the Faculty of Electrical Engineering, Czech Technical University, where he is currently a Professor of radio engineering. He has led a number of industrial and research projects financed by the European Union and national agencies. His research interests include wireless communication and information theory, cooperative and distributed modulation, wireless network coding and distributed signal processing, multiple-input–multiple-output systems, nonlinear space–time modulation and coding, and iterative processing. Dr. Sykora has served on various IEEE conferences as Technical Program and Organizing Committee member and chair.

Wai Ho Mow (S’89–M’93–SM’99) received his M.Phil. and Ph.D. degrees in information engineering from the Chinese University of Hong Kong in 1991 and 1993, respectively. From 1997 to 1999, he was with the Nanyang Technological University, Singapore. He has been with the Hong Kong University of Science and Technology (HKUST) since March 2000. He was the recipient of seven research/exchange fellowships from five countries, including the Humboldt Research Fellowship. His research interests are in the areas of communications, coding, and information theory. He pioneered the lattice approach to signal detection problems (such as sphere decoding and complex lattice reduction-aided detection) and unified all known constructions of perfect roots-of-unity (aka CAZAC) sequences (widely used as preambles and sounding sequences). He has published one book, and has co-authored over 30 filed patent applications and over 160 technical publications, among which he is the sole author of over 40. He co-authored two papers that received the ISITA2002 Paper Award for Young Researchers and the APCC2013 Best Paper Award. He is currently the leader of the HKUST Barcode Group which won the Best Mobile App Award at ACM MobiCom’2013 by developing a novel picture-embedding barcode app, called PiCode. Since 2002, he has been the principal investigator of 16 funded research projects. In 2005, he chaired the Hong Kong Chapter of the IEEE Information Theory Society. He was the Technical Program Co-Chair of five conferences, and served the technical program committees of numerous conferences, such as ICC, Globecom, ITW, ISITA, VTC and APCC. He was a Guest Associate Editor for numerous special issues of the IEICE Transactions on Fundamentals. He was an industrial consultant for Huawei, ZTE, and Magnotech Ltd. He was a member of the Radio Spectrum Advisory Committee, Office of the Telecommunications Authority, Hong Kong S.A.R. Government from 2003 to 2008.